Welcome to Calculus 1000A 002!

Class info

  • Instructor name: Pál Zsámboki
  • Instructor email: pzsambok@uwo.ca
  • Instructor office: MC 133
  • Class webpage: on OWL
  • Documents on OWL page:
    • Class Outline written by the Course Coordinator
    • List of practice problems
  • Textbook: Single Variable Calculus: Early Transcendentals (8th edition), by James Stewart (Brooks/Cole)
    • It is important that you have access to the correct edition, since there is a list of practice problems we'll suppose you know how to solve.
  • Office hours: Thursdays 3pm-5pm. No office hour on September 23.
  • Math Help Centre: TBD, starting on September 19

Course grade components

  • Midterm exam 35%
    • The midterm exam will be on Friday October 21, from 7pm-9pm. Location TBA. The exam will be closed book, no notes, no calculators or any other electronic devices will be permitted.
  • Final exam 50%
    • The final exam will be cumultative, and it will be 3 hours long. Location TBA. The exam will be closed book, no notes, no calculators or any other electronic devices will be permitted.
  • Quizzes 15%
    • There will be 6 online quizzes on OWL, with 2 questions in each.
    • You will have 2 weeks for every one of them, and you'll have 3 tries with each.
    • Each quiz will have a time limit of 20 minutes.
    • Your worst quiz grade will be dropped.
    • For more details, check quizzes.txt.

Trigonometry

Parametrization of the unit circle

  • I think the most intuitive way of thinking about trigonometric functions is that the function \(f(\theta)=(\cos(\theta),\sin(\theta))\) is the parametrization of the unit circle with respect to arc length, in counterclockwise or in other words positive direction, starting from the point \(f(0)=(1,0)\).
    • That is, the function \(f(\theta)\) traces out the unit circle as \(\theta\) varies
    • Being parametrized with respect to arc length means that if we change \(\theta\) by 1 unit, the parametrization \(f(\theta)\) traces out an arc of length 1.

Standard position, positive and negative angles

  • The standard parametrization \(f(\theta)=(\cos(\theta),\sin(\theta))\) corresponds to the following conventions.
    • An angle is said to be in standard position, if its vertex is placed at the origin of the coordinate system we're using, and its initial side is on the positive \(x\)-axis
    • A positive angle is obtained if going from the initial angle to the terminal angle is a counterclockwise rotation
    • A negative angle is obtained if going from the initial angle to the terminal angle is a clockwise rotation

Circles and arcs of general radii

  • If the angle \(\theta\) is measured like this, that is in the way that the parametrization of the unit circle \(f(\theta)=(\cos(\theta),\sin(\theta))\) is with respect to arc length, we say that it is measured in radians. Since the circumference of the unit circle is \(2\pi\), a complete revolution is done by changing \(\theta\) by \(2\pi\).
    • Let \(r>0\) be a positive number. Then similarly we can parametrize the circle \(C\) with centre the origin and radius \(r\) via \(f(\theta)=(r\cos(\theta),r\sin(\theta))\).
    • Note that this parametrization is not with respect to arc length: a complete revolution around \(C\), which traces out a path of length the circumference of \(C\), that is \(a=2\pi r\), corresponds to a change of \(2\pi\) in \(\theta\).
    • More generally, for a circular arc of length \(a\), radius \(r\), and angle \(\theta\), we get the formula \(a=\theta r\).

Degrees

  • There is another, historical angle unit, that of degrees. If angles are measured in degrees, then a complete revolution is \(360^\circ\).
    • This gives us the conversion formula between radians and degrees: \(2\pi=360^\circ\).
    • A related unit of length which I'm rather fond of is that of nautical miles (NM). 1 nautical mile is 1 latidual degree minute. That is, if you travel 60 NM from North to South, then you travelled precisely \(\tfrac{1}{360}\) of the latitudal circumference of the Earth.
    • Exercises: D.2,4,8,10

Trigonometric functions

  • Consider a right triangle \(T\), where one of the non-right angles is \(\theta\).
    • Suppose that the angle \(\theta\) is in standard position, let's give it the positive orientation, and let's denote the length of the hypotenuse by \(r\).
    • Then we can study the triangle \(T\) with the parametrization \(f(\theta)=(x=r\cos(\theta),y=r\sin(\theta))\).
    • Its vertices are going to be \(O(0,0)\), \(X(x,0)\), and \(P(x,y)\).
    • The side \(\overline{OX}\) is the adjacent, the side \(\overline{XP}\) is the opposite, and the side \(\overline{OP}\) is the hypotenuse.
    • The lengths of the legs are \(|OX|=x\), and \(|XP|=y\).
    • Then we can express the trigonometric functions using these side lengths:
    • \(\cos(\theta)=\frac{x}{r}\), \(\sin(\theta)=\frac{y}{r}\), \(\tan(\theta)=\frac{y}{x}=\frac{\sin(\theta)}{\cos(\theta)}\),
    • \(\sec(\theta)=\frac{r}{x}=\frac{1}{\cos(\theta)}\), \(\csc(\theta)=\frac{r}{y}=\frac{1}{\sin(\theta)}\), \(\cot(\theta)=\frac{x}{y}=\frac{\cos(\theta)}{\sin(\theta)}\)
    • Since the parametrization \(f(\theta)\) is defined for all values of \(\theta\), we can use the same formulas in case \(\theta<0\) or \(\theta\ge\frac{\pi}{2}\).

Graphs of the trigonometric functions

  • Note the following properties on the graphs of \(\cos(x)\) and \(\sin(x)\).
    • Both functions are \(2\pi\)-periodic, their domain is \((-\infty,\infty)\), and their range is \([-1,1]\).
    • \(\cos(x)=\sin(x+\frac{\pi}{2})\). This can be also seen by rotating the right triangle \(OXP\) by \(\frac{\pi}{2}\).
    • \(\cos(x)=0\) iff (if and only if) \(x=\frac{\pi}{2}+k\pi\) for some integer \(k\), and \(\sin(x)=0\) iff \(x=k\pi\).
    • I think it makes remembering the special values easier to notice that on the interval \([0,\frac{\pi}{2}]\), the functions \(\cos(x)\) is decreasing, and the function \(\sin(x)\) is increasing.